A probabilistic automata network epidemic model with births and deaths exhibiting cyclic behaviour

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Abstract
A probabilistic automata network model for the spread of an infectious disease in a population of moving individuals is studied. The local rule consists of two subrules. The first one, applied synchronously, models infection, birth and death processes. It is a probabilistic cellular automaton rule. The second, applied sequentially, describes the motion of the individuals. The model contains six parameters: the probabilities p for a susceptible to become infected by contact with an infective; the respective birth rates bs, and bi of the susceptibles from either a susceptible or an infective parent; the respective death rates ds, and di of susceptibles and infectives; and a parameter m characterizing the motion of the individuals. The model has three fixed points. The first is trivial, it describes a stationary state with no living individuals. The second corresponds to a disease-free state with no infectives. The third and last one characterizes an endemic state with non-zero densities of susceptibles and infectives. Moreover, the model may exhibit oscillatory behaviour of the susceptible and infective densities as functions of time through a Hopf-type bifurcation. The influence of the different parameters on the stability of all these states is studied with a particular emphasis on the influence of motion which has been found to be a stabilizing factor of the cyclic behaviour.

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